How do you evaluate #\log _ { 3} 2+ 3\log _ { 3} 4#?

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Answer 1 · 2017-08-06T09:58:11

#4.4165#

If you are adding logs with the same base, it it the same as multiplying the numbers.

#log_3 2 + 3log_3 4" "larr# apply the power law.

#=log_3 2 + log_3 4^3#

#=log_3 (2 xx 64)#

#log_3 128#

#128# is not a power of #3#, so to continue requires a calculator.
We can estimate the answer:

#3^4 = 81 and 3^5 = 243#, so the value has to be between #4 and 5#

Using the change of base law:

#(log_10 128)/(log_10 3)#

#=4.4165#